Ch 2 · 5.3 Circles, Arcs and Sectors · O Level Mathematics 4024
Topic 2 of 3 in Mensuration — O Level Mathematics 4024 — 18 lessons.
Introduction
Wheels, pizzas and clock faces are circles; a slice of pizza is a sector and its crust is an arc. A sector is simply a fraction of the whole circle, so if you know the circle formulas you can find any arc or sector. (Syllabus section 5.3.)
Key ideas
C = 2π r = π d A = π r^2
🧑🏫 Teacher's explanation
Think of a circle as a whole pizza. The circumference is the crust around the edge, and the area is the total amount of cheese covering the top. If you only eat one slice, that slice is a 'sector'. The crust of that specific slice is the 'arc length'.
📖 Definitions
🔍 Worked example — Easy
A circle has a radius of 7 cm. Calculate the circumference and the area. Give your answers in terms of π.
🔍 Worked example — Standard
A sector of a circle has a radius of 10 cm and an angle of 72^° at the center. Calculate the arc length and the sector area. Give your answers in terms of π.
🔍 Worked example — Exam-style
A major sector has an arc length of 25 cm and a radius of 5 cm. Calculate the angle of the sector in degrees. Give your answer to 1 decimal place.
⚠️ Common errors
🧠 Quick check 1
🧠 Quick check 2
🎯 Exam question
Worked examples
Radius 7 cm: circumference = 14π = 44.0 cm; area = 49π = 154 cm².
🔍 Worked example — Easy
A circle has a radius of 7 cm. Calculate its circumference and area. Give your answers in terms of π and rounded to 1 decimal place.
🔍 Worked example — Standard
A sector of a circle has a radius of 10 cm and a central angle of 72^°. Find the arc length and the area of the major sector. Give answers in terms of π.
🔍 Worked example — Exam-style
A sector of a circle has a radius of 6 cm and an arc length of 11 cm. Find the angle of the sector, θ, to 1 decimal place.
⚠️ Common errors
Common mistakes
Practice
Answers: (1) 36π cm². (2) arc 2π cm; area 9π cm². (3) r = 50/2π = 7.96 cm. (4) θ/360 × 64π = 16π ⇒ θ = 90^°.